12.13 Velocity Selector

Introduction

A velocity selector is a device that uses perpendicular electric and magnetic fields to filter a beam of charged particles, allowing only those with a specific velocity to pass through undeflected. This method demonstrates the interplay between electric and magnetic forces on a moving charge and is a critical component in instruments such as mass spectrometers.

Configuration of Electric and Magnetic Fields

A velocity selector is constructed by applying a uniform electric field ($\vec{E}$) and a uniform magnetic field ($\vec{B}$) that are perpendicular to each other and also perpendicular to the initial velocity ($\vec{v}$) of the charged particles. This arrangement is called crossed fields.

A charged particle entering this region experiences two distinct forces:

• Electric Force ($\overrightarrow{F_E}$): Acts in the direction of the electric field (for a positive charge): $\overrightarrow{F_E}=q\vec{E}$
• Magnetic Force ($\overrightarrow{F_B}$): Acts perpendicular to both the velocity and the magnetic field (right-hand rule), in the opposite direction to the electric force: $\overrightarrow{F_B}=q(\vec{v}\times \vec{B})$

The Lorentz Force

The total force on a charged particle moving through both electric and magnetic fields is the Lorentz Force — the vector sum of the electric and magnetic forces: $\vec{F}=q\vec{E}+q(\vec{v}\times \vec{B})$

Condition for No Deflection

For a particle to pass through undeflected, the net force must be zero — the electric and magnetic forces must be equal in magnitude and opposite in direction: $F_E=F_B$ $qE=qvB$

Cancelling $q$ from both sides gives the selected velocity: $\boxed{v=\frac{E}{B}}$

Only particles travelling at precisely this speed pass straight through. This result is independent of the particle's charge ($q$) and mass ($m$).

Dimensions→

How the Filtering Works

Particles with velocities different from $v=E/B$ are deflected and blocked by slits:

Key Equations Summary

Significance

The velocity selector is most notably used as the first stage of a mass spectrometer, ensuring all ions entering the analysis chamber have the same known velocity $v=E/B$. This allows subsequent separation of ions to be based purely on their mass-to-charge ratio.

Note: Reversing the direction of $\vec{B}$ would cause both forces to act in the same direction — no particle could pass undeflected regardless of its speed.